Monte Carlo Analysis of the Effects of Varying the Number of Steady State Serum Levels on Vancomycin AUC Calculations with the Goti Two-Compartment Model Compared to a One-Compartment Model

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This study evaluates the impact of the number of serum levels on the accuracy of vancomycin AUC calculations using one- and two-compartment pharmacokinetic models. Two-compartment model dosing methods are commonly recommended for vancomycin dosing. The number of levels required to adequately describe a two-compartment model is reported to be four to eight levels. A pharmacokinetic modeling analysis of vancomycin was conducted using the Monte Carlo method in Excel. The Monte Carlo method is a technique of random sampling employed to approximate solutions to quantitative problems. It is a mathematical procedure or algorithm in which random numbers are run through a model or simulation to observe the properties of large sets of results.

The Monte Carlo method was used to determine the effects of varying the number of steady state serum levels on AUC calculations and pharmacokinetic parameter optimization using the Goti two-compartment model as compared to a one-compartment model.  Monte Carlo simulations evaluated the two-compartment model using the following post-dose level comparison comparisons: eight levels (0, 1, 2, 3, 4, 8, 9, and 10 hours); six levels (1, 2, 3, 4, 8, and 10 hours); four levels (2, 4, 8, and 10 hours); two levels (2, and 10 hours); and one level (10 hours). The Goti model's population parameter mean values were used to calculate post-dose levels for a Q12H regimen with a two hour infusion period, employing standard two-compartment intermittent infusion open model equations. These levels were considered to be the actual or true levels. A data set of 1,000 rows with randomized values for each true level were created to mimic the normal assay uncertainty in measured levels. The same randomized levels were used in each simulation, so the simulations could be compared to one another. The randomized levels represented the measured levels.

The randomized values followed a normal distribution with a mean equal to the actual level and standard deviation of a typical assay assuming a coefficient of variation of 0.1. This was done by setting cells in 1,000 rows in Excel = NORMINV(rand(), actual level, standard deviation of level) for each actual level which produced random values that fit a normal distribution described by the input actual level and standard deviation of the assay. Rand() is an Excel function that produces random numbers greater than or equal to 0 and less than 1. It does not require an argument. Excel's NORMINV(probability, mean, standard deviation) function requires the arguments in parenthesis. It returns a value described by its probability in a normal distribution for the input mean and standard deviation. If all 1,000 randomized values for each true level were plotted on a histogram a normal curve would result with the input mean and standard deviation.

Data fittings were performed for each row using Excel Solver's Revolutionary routine to minimize the sum of the square of errors for the randomized values of the actual levels versus the predicted levels for the two-compartment model by optimizing the pharmacokinetic parameters (Vd central, Cl total, Vd peripheral, and Cl distribution).  During data fitting, standard two-compartment intermittent infusion open model equations were used to calculate the predicted levels, and AUC at steady state for the fit Vd central, Vd peripheral, Cl total, and Cl distribution of each row. Standard one-compartment intermittent infusion open model equations were used to calculated Vd, Clearance, and AUC using the randomized levels at two and ten hours post-dose, as this is typically done in clinical practice.  For the Monte Carlo simulation with a single 10 hour post-dose level, the Vd for the one-compartment model was held at the mean value previously calculated using the 2 hour and 10 hour post-dose randomized levels. Clearance was then calculated using Excel's solver by minimizing the square of the error for (randomize trough  - predicted trough level)².

 

One-Compartment Model Intermittent Infusion

K_1/hours = ln(Cpmax_mg/L/Cpmin_mg/L) / Time between levels_hours

K _1/hours = Clearance_L/h / Vd_L

Cp_mg/L steady state= Dose_mg*(1-exp(-K*Infusion Period_hours)) * exp(-K* Time of level post-dose_hours) / (Vd_L*K*Infusion Period_hours *(1-exp(-K*Tau_hours)))

Vd_L = Dose_mg*(1-exp(-K*Infusion Period_hours)) * exp(-K* Time of level post-dose_hours)  / (Cp_mg/L*K*Infusion Period_hours *(1-exp(-K*Tau_hours)))

AUC_mg*h/L = Dose_mg*(24/Tau_Hours) / Clearance_L/h

Two-Compartment Model Intermittent Infusion

Cp = [Dose*(K21-alpha)*(1-exp(-alpha*Infusion Period))*exp(-alpha*Time post Infusion) / (Infusion Period*Vcentral*alpha*(beta-alpha)*(1-exp(-alpha*Tau)))] +

         Dose*(Beta-K21)*(1-exp(-beta*Infusion Period))*exp(-beta*Time post Infusion) / (Infusion Period*Vcentral*beta*(beta-alpha)*(1-exp(-beta*Tau)))

AUC_mg*h/L= Dose_mg*(24/Tau_Hours) / Clearance total_L/h

Objective Equation To Minimize for each two-compartment model simulations

Minimize the sum of the following for all levels drawn during the simulation = (randomized value of true level_1-n - predicted level_1-n)²

Patient Demographics

Weight 70 kg

Creatinine Clearance 120 ml/min

Goti Two-Compartment Model Parameters

Cl L/h total population mean= 4.5 * (creatinine clearance/120)^0.8 , Vd L central population mean = 58.4 *(weight_kg / 70), Vdperipheral 38.4 L, Cl distribution 6.5 L/h

 Coefficient of variation for Vd central, Vd peripheral, Cl total and Cl distribution were set at 0.3 . Coefficient of variation of lab assay was set at 0.1.

Other Equations and Values Used in the Analysis

CV = Standard Deviation of Parameter / Mean

Dose (mg) = 1012 (dose chosen to give an AUC of 450 for population mean Goti Parameters), Tau = 12, Infusion Period = 2 hour

Results of Analysis and Recommendations

Accuracy of  AUC calculation using a two-compartment model increased as the number of levels measured in alpha and beta phases increased as expected. When comparing the percentage of AUCs calculated within +/- 50 of the actual AUC when only a trough was measured the one-compartment model was similar (69.5%) to the two-compartment model (62.3%). When comparing the percentage of AUCs calculated within +/- 50 of the actual AUC the one-compartment model with two measured levels (82%) was significantly better than the two-compartment model with one measured level (62%), was slightly better than the two-compartment model with two measured levels (79.7%), was slightly worse than the two-compartment model with four measured levels (87.8%), was significantly worse than the two-compartment model with eight measured levels (97.8%).

As the number of measured levels increased the standard deviation of the fit clearance decreased for the two-compartment model: 1 level SD 0.62, 2 levels SD 0.4, 4 levels SD 0.36, and 8 levels SD 0.23; reducing the variability of the fit clearance and the AUC calculation error. There was an inverse linear correlation between the clearance SD and AUC calculation accuracy R² 0.9891.

As the number of measured levels increased the standard deviation of the fit Vd central decreased for the two-compartment model: 1 level SD 26.6, 2 levels SD 25.07, 4 levels SD 24.06, 8 levels SD 15.3.

Unless four or more levels were drawn,  the two-compartment model AUC calculations were not more accurate than a one-compartment model with two levels. Two-compartment AUC calculations with eight levels were very accurate.  This is an indication that the two-compartment model was over parameterized without an adequate number of measured levels.

A one or two-compartment model with a trough level is not recommended for AUC dosing due to the high rate of AUC calculation errors.

This is not a Bayesian analysis, but was an attempt to quantify the effects of the number of measured levels on one and two-compartment model AUC calculations using the Goti model as an example. The analysis does indicate the need to draw an adequate number of levels when building a two-compartment model if accuracy is desired. Also an adequate number of levels are required if the data will be the basis of a Bayesian model to accurately determine the SD of pharmacokinetic parameters.

 

	Percentage of AUCs in Specified Ranges (mg*h/L)
	300-399	400-500	501-599	600-699
	Two-Compartment Model
Number of Levels
1	17.30	62.33	16.89	3.48
2	7.51	79.72	12.07	0.71
4	2.13	87.75	9.21	0.91
8	1.11	97.78	1.11	0.00
	Percentage of AUCs in Specified Ranges (mg*h/L)
	300-399	400-500	501-599	600-699
	One-Compartment Model
2	3.76	82.01	14.23	0.00
1	30.09	69.50	0.41	0.00

 

		Two-Compartment Model
	Average Fit Pharmacokinetic Parameters & Standard Deviation
	Vd central L	Clearance Total L/h	Vd Peripheral L	Cl distribution L/h	AUC mg*h/L
True Parameters	58.4	4.5	38.4	6.5	450
Number of Levels
1	64.12	4.52	38.73	6.60	458.00
SD	26.64	0.62	18.44	2.84	71.08
2	61.39	4.49	38.92	6.74	455.27
SD	25.07	0.40	18.09	2.80	42.13
4	60.71	4.46	39.67	6.67	457.09
SD	24.06	0.36	18.36	2.71	39.49
8	62.89	4.52	37.83	6.64	448.88
SD	15.30	0.23	17.88	2.78	22.40
		One-Compartment Model
	Average Fit Pharmacokinetic Parameters & Standard Deviation
	Vd L	Clearance L/h			AUC  mg*h/L
2	100.02	4.42			461.24
SD	54.75	0.35			35.32
1	100.02	4.91			415.08
SD	VD fixed Pop Mean	0.41			33.58

 

Future Work

A Bayesian analysis will be performed in the future to see if AUC calculations are improved as compared to the non-Bayesian method.

Bayesian Equation Two-Compartment Model

Minimize the sum of the following = (measured level_1-n - predicted level_1-n)² / (0.1 * measured level_1-n)²  +  (Vd central population mean - Vd central fit)² / (0.3*Vd central population mean)²  +  (Cl total population mean - Cl total fit)² / (0.3*Cl total population mean)² + (Vd peripheral population mean - Vd peripheral fit)² / (0.3*Vd peripheral population mean)²  +  (Cl distribution population mean -Cl distribution fit)² / (0.3*Cl distribution population mean)²

 

References
1. Goti V, Chaturvedula A, Fossler MJ, Mok S, Jacob JT. Hospitalized Patients With and Without Hemodialysis Have Markedly Different Vancomycin Pharmacokinetics: A Population Pharmacokinetic Model-Based Analysis. Ther Drug Monit. 2018 Apr;40(2):212-221. doi: 10.1097/FTD.0000000000000490. Erratum in: Ther Drug Monit. 2019 Aug;41(4):549. doi: 10.1097/FTD.0000000000000666. PMID: 29470227.

2. Basic Pharmacokinetics and Pharmacodynamics: An Integrated Textbook and Computer Simulations 2nd Edition Wiley; 2016
by Sara E. Rosenbaum (Editor)

3. Excel for Scientists and Engineers Numerical Methods by E. Joseph Billo